Sufficient Condition on the existence of an intersection between N spheres

I have this problem to solve:

Given N spheres with centers C : {$$c_1, ..., c_n| c_i ∈ R^M$$} and radiuses R : {$$r_1, ..., r_n| r_i ∈ R$$}

I'm searching for a sufficient condition which states that exists a region that is part of each sphere.

For N = 2 the condition is simple: if the euclidean distance d($$c_1, c_2$$) < $$r_1 + r_2$$ then the region exists

For N > 2 I have difficult both finding a sufficient condition myself or finding it online

Can someone help me?

• Welcome to MSE. In order for MathJax commands to be effective, they must be surrounded by $ signs. For example, $x^2$ shows up as$x^2$. Apr 9, 2021 at 14:13 • Thanks, I didn't knew the syntax Apr 9, 2021 at 14:15 • By Helly's theorem, in the plane it's enough to do the$N=3$case, and in$\mathbb{R}^n$it's enough to do the$N=n+1\$ case. Even the case of three disks in the plane doesn't seem to be easy. I found this paper on computing the area of the overlap. It might have something useful in it. Apr 9, 2021 at 14:46